Duality for Landau-Ginzburg models

Claude Sabbah

arxiv: 2212.07745 · v3 · submitted 2022-12-15 · 🧮 math.AG

Duality for Landau-Ginzburg models

Claude Sabbah This is my paper

Pith reviewed 2026-05-24 10:30 UTC · model grok-4.3

classification 🧮 math.AG

keywords dualityLandau-Ginzburg modelsquasi-projective varietyregular functionalgebraic geometrycomplex geometry

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The pith

Duality statements attach to pairs of smooth quasi-projective varieties and regular functions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper surveys various duality statements associated with a smooth complex quasi-projective variety together with a regular function on it. These statements arise in the setting of Landau-Ginzburg models. A sympathetic reader cares because the statements relate different geometric and analytic features of the same pair. The survey organizes and presents results drawn from the existing literature on the subject.

Core claim

Various duality statements are attached to a pair consisting of a smooth complex quasi-projective variety and a regular function on it.

What carries the argument

The pair consisting of a smooth complex quasi-projective variety and a regular function on it, to which the duality statements are attached.

If this is right

The dualities relate different invariants attached to the pair.
The statements apply within the theory of Landau-Ginzburg models.
The dualities connect algebraic and analytic structures on the given data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The organization may make it easier to identify which duality applies to a concrete computation.
The statements could serve as a starting point for extending duality results beyond the quasi-projective smooth case.

Load-bearing premise

The survey accurately represents the cited duality statements from the prior literature without introducing errors in selection or presentation.

What would settle it

Discovery of a misstatement or inaccurate selection in the description of any cited duality would show that the survey does not correctly represent the literature.

read the original abstract

This article surveys various duality statements attached to a pair consisting of a smooth complex quasi-projective variety and a regular function on it. It is dedicated to the memory of Bumsig Kim.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a survey compiling known duality statements for Landau-Ginzburg models with no new results or proofs.

read the letter

This paper surveys duality statements attached to pairs (X, f) where X is a smooth complex quasi-projective variety and f is a regular function. It adds nothing new to the literature and makes no original claims or derivations. The main service is pulling together existing results on these dualities into one document, which could help readers track down statements scattered across papers on Landau-Ginzburg models and mirror symmetry. That organization is the only concrete contribution here. The value depends on whether the selection of statements is representative and whether the citations are presented without distortion. Since the work contains no proofs or calculations to check, there is no internal mathematical error to flag, but the survey format means any gaps in coverage or emphasis would be the main risk. The dedication to Bumsig Kim suggests the author aimed for a careful overview rather than advocacy for one approach. This is useful for specialists already active in algebraic geometry who want a consolidated reference list rather than a new framework. A reader seeking fresh theorems or resolved open questions will find none. I would send it for peer review because a well-executed survey of this type can still save time for the community, even if it does not advance the frontier.

Referee Report

0 major / 0 minor

Summary. This article surveys various duality statements attached to a pair consisting of a smooth complex quasi-projective variety and a regular function on it. It is dedicated to the memory of Bumsig Kim.

Significance. As a survey paper with no original theorems or derivations claimed, its value lies in compiling and organizing existing duality results from the literature on Landau-Ginzburg models. If the citations and presentations are accurate, it may serve as a useful reference for researchers in algebraic geometry and mirror symmetry.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of our survey article. We are pleased that the manuscript is viewed as a useful compilation of duality results for Landau-Ginzburg models.

Circularity Check

0 steps flagged

Survey with no derivations or fitted quantities

full rationale

The paper is explicitly a survey attaching existing duality statements to pairs (X, f) where X is smooth complex quasi-projective and f regular; no original theorems, proofs, or derivations are claimed. The reader's weakest assumption (faithful representation of prior literature) is the only potential point of failure, but the provided abstract and description contain no internal mathematical construction whose assumptions could be inconsistent or under-supported. Without an original argument, there is no load-bearing technical condition to test for circularity. This is the most common honest finding for survey papers.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As a survey paper, no new free parameters, axioms, or invented entities are introduced by the authors; the contribution is compilation of existing material.

pith-pipeline@v0.9.0 · 5529 in / 852 out tokens · 16873 ms · 2026-05-24T10:30:36.608293+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem A. The C[u]-modules H^k(Y,(Ω^•_Y(log H)[u],ud+df)) and H^k(Y,(Ω^•_Y(log H)(-H)[u],ud-df)) are C[u]-free of finite rank, and equipped with a meromorphic connection having a pole of order at most 2 at u=0...
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 2.3 (J.-D. Yu). The corresponding cohomological pairing H^{n+k}(X,(Ω^•_f,d+df)) ⊗ H^{n-k}(X,(Ω^•_f(-D),d-df)) → H^{2n}_dR(X) is perfect.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.